Finite Group Databasebeta

← #13328645

Subgroups — #13328645

Order 1792.

998 conjugacy classes of subgroups; only the 207 normal subgroups are shown.

normal subgroups only

This group has 998 conjugacy classes of subgroups — too many to draw. Only its 207 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3502C211yesmaximal, characteristic, complemented
2#11740800C212yesmaximal
3#11739921C211yesmaximal, characteristic
4#11729656C218yesmaximal, complemented
5#11729589C212yesmaximal, complemented
6#11729378C211yesmaximal, characteristic, complemented
7C7×Q82—22no—
8C7×#557—22no—
9C7×#557—24no—
10C7×#544—21no—
11C7×#544C2214yes—
12C7×#533—22no—
13C7×#533C2212yes—
14C7×#519—21no—
15C2×C4×C7×Q8C2211yescharacteristic
16C2×#6510—21no—
17C2×#6437C2211yescharacteristic
18#89860—22no—
19#89857—22no—
20#89818C2218yes—
21#89799—24no—
22#89763C2214yes—
23#89760—21no—
24#89711—22no—
25C4×C4⋊Dic7C2211yescharacteristic
26#89697C2212yes—
27#89333C2218yes—
28#89323C2214yes—
29#18403—71nomaximal, Sylow
30C22×C7×Q8D411yescharacteristic
31C22×D28—21no—
32C22×D28—21no—
33C22×D28C2312yes—
34C7×C4⋊Q8—22no—
35C7×C4⋊Q8—22no—
36C7×C4⋊Q8—24no—
37C7×C4⋊Q8—28no—
38C7×C4⋊Q8D412yes—
39C7×C4⋊Q8D412yes—
40C7×#125—24no—
41C7×#125—24no—
42C7×#125—28no—
43C7×#125—22no—
44C7×C22⋊Q8—22no—
45C7×C22⋊Q8—24no—
46C7×C22⋊Q8C2314yes—
47C4×C7×Q8—24no—
48C4×C7×Q8—24no—
49C4×C7×Q8—24no—
50C4×C7×Q8—24no—
51C4×C7×Q8—22no—
52C4×C7×Q8—22no—
53C4×C7×Q8—22no—
54C4×C7×Q8D414yes—
55C7×#117—24no—
56C7×#117—22no—
57C2×C7×C4⋊C4D411yescharacteristic
58C2×C7×C4⋊C4C2312yes—
59C2×C42×C7D412yes—
60C2×C42×C7C2311yescharacteristic
61#6510—44no—
62C2×#1143—28no—
63C2×(C2×D7)⋊C4—24no—
64C2×(C2×D7)⋊C4—24no—
65C2×C4⋊Dic7—21no—
66C2×C4×Dic7—21no—
67#6437—22no—
68#6437—22no—
69C7⋊#95C2312yes—
70C22⋊(C7⋊C8)C2314yes—
71#6383—28no—
72C4⋊(C7⋊C8)—22no—
73C4×C7⋊C8—22no—
74#3502D711yescharacteristic, complemented
75#3208—72no—
76#2548—71no—
77#1599—78no—
78#1503—72no—
79#1461—71no—
80C23×D7—22no—
81C2×C7×Q8—24no—
82C2×C7×Q8—22no—
83C2×C7×Q8—22no—
84C2×C7×Q8—22no—
85C2×C7×Q8SD1614yescomplemented
86C2×C7×Q8C2×D414yes—
87C22×C4×C7C2×D411yescharacteristic
88C22×C4×C7C2×D412yes—
89C22×C4×C7C2×D412yes—
90C22×C4×C7C2×D411yescharacteristic
91C22×C4×C7C2411yescharacteristic
92C22×Dic7—22no—
93C2×D28—24no—
94C2×D28—44no—
95C2×D28—41no—
96C2×D28—42no—
97C2×D28—41no—
98C2×D28—42no—
99C2×D28—42no—
100C2×D28—82no—
101C7×C4⋊C4—24no—
102C7×C4⋊C4—24no—
103C7×C4⋊C4—24no—
104C7×C4⋊C4—28no—
105C7×C4⋊C4—22no—
106C7×C4⋊C4—22no—
107C7×C4⋊C4—22no—
108C7×C4⋊C4—22no—
109C7×C4⋊C4—24no—
110C7×C4⋊C4—24no—
111C7×C4⋊C4—24no—
112C7×C4⋊C4C2×D414yes—
113C7×C22⋊C4—24no—
114C42×C7—28no—
115C42×C7—22no—
116C42×C7—22no—
117C42×C7—24no—
118C42×C7C2×D412yes—
119C42×C7C2×D412yes—
120#1143—88no—
121(C2×D7)⋊C4—48no—
122(C2×D7)⋊C4—84no—
123C4⋊Dic7—42no—
124C4×Dic7—42no—
125C2×C7⋊C8—24no—
126Q82—22no—
127#557—22no—
128#557—24no—
129#544—21no—
130#544C2×D714yes—
131#533—22no—
132#533C2×D712yes—
133C2×#127—71no—
134C2×#124—141no—
135#519—21no—
136C2×C4×Q8C2×D711yescharacteristic
137#492—142no—
138#486—142no—
139#460—144no—
140#450—78no—
141#431—72no—
142#394—74no—
143#390—141no—
144#390—142no—
145C4×C4⋊C4—71no—
146#328—78no—
147#324—74no—
148C23×C7C22×D411yescharacteristic
149C22×D7—44no—
150C22×D7—44no—
151C22×D7—42no—
152C22×D7—42no—
153C22×D7—42no—
154C7×Q8—28no—
155C7×Q8—44no—
156C7×Q8—44no—
157C7×Q8—84no—
158C2×C4×C7—22no—
159C2×C4×C7—22no—
160C2×C4×C7—22no—
161C2×C4×C7—24no—
162C2×C4×C7—24no—
163C2×C4×C7—22no—
164C2×C4×C7—41no—
165C2×C4×C7—42no—
166C2×C4×C7#12011yescharacteristic
167C2×C4×C7#12012yes—
168C2×C4×C7#12011yescharacteristic
169C2×C4×C7C2×SD1614yes—
170C2×C4×C7C2×SD1612yes—
171C2×C4×C7#13712yes—
172C2×C4×C7C22×D412yes—
173C2×Dic7—42no—
174C2×Dic7—42no—
175C2×Dic7—42no—
176D28—48no—
177D28—82no—
178D28—84no—
179D28—82no—
180C7⋊C8—44no—
181C22×Q8C22⋊D711yescharacteristic
182C22×D4—72no—
183C22×D4—141no—
184C22×D4—141no—
185C2×SD16—148no—
186C4⋊Q8—22no—
187C4⋊Q8—22no—
188C4⋊Q8—24no—
189C4⋊Q8—28no—
190C4⋊Q8C22⋊D712yes—
191C4⋊Q8C22⋊D712yes—
192#127—142no—
193#127—142no—
194#125—24no—
195#125—24no—
196#125—28no—
197#125—22no—
198#124—284no—
199C22⋊Q8—22no—
200C22⋊Q8—24no—
201C22⋊Q8C22×D714yes—
202C4×Q8—24no—
203C4×Q8—24no—
204C4×Q8—24no—
205C4×Q8—24no—
206C4×Q8—22no—
207C4×Q8—22no—
208C4×Q8—22no—
209C4×Q8C22⋊D714yes—
210#117—24no—
211#117—22no—
212C2×C4⋊C4—141no—
213C2×C4⋊C4C22⋊D711yescharacteristic
214C2×C4⋊C4C22×D712yes—
215C2×C22⋊C4—144no—
216C2×C22⋊C4—144no—
217C2×C42—141no—
218C2×C42C22⋊D712yes—
219C2×C42C22×D711yescharacteristic
220C4⋊C8—142no—
221#102—148no—
222C22⋊C8—74no—
223C4×C8—142no—
224#95—72no—
225C22×C7—21no—
226C22×C7—21no—
227C22×C7C2×#12011yescharacteristic
228C22×C7C22×SD1611yescharacteristic
229C22×C7C2×#13711yescharacteristic
230C2×D7—84no—
231C2×D7—84no—
232C2×D7—82no—
233C2×D7—82no—
234C2×D7—82no—
235C4×C7—24no—
236C4×C7—21no—
237C4×C7—22no—
238C4×C7—44no—
239C4×C7—42no—
240C4×C7—44no—
241C4×C7#44814yes—
242C4×C7#45412yes—
243Dic7—82no—
244C24—142no—
245C2×Q8—24no—
246C2×Q8—22no—
247C2×Q8—22no—
248C2×Q8—22no—
249C2×Q8#114314yescomplemented
250C2×Q8C2×C22⋊D714yes—
251C2×D4—144no—
252C2×D4—284no—
253C2×D4—282no—
254C2×D4—282no—
255C2×D4—281no—
256C2×D4—282no—
257C2×D4—281no—
258C2×D4—562no—
259C22×C4—142no—
260C22×C4C2×C22⋊D712yes—
261C22×C4C2×C22⋊D711yescharacteristic
262C22×C4C2×C22⋊D712yes—
263C22×C4C2×C22⋊D711yescharacteristic
264C22×C4C23×D711yescharacteristic
265SD16—568no—
266C2×C8—144no—
267C4⋊C4—24no—
268C4⋊C4—24no—
269C4⋊C4—24no—
270C4⋊C4—28no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—24no—
276C4⋊C4—24no—
277C4⋊C4—24no—
278C4⋊C4—282no—
279C4⋊C4C2×C22⋊D714yes—
280C22⋊C4—24no—
281C22⋊C4—288no—
282C22⋊C4—564no—
283C42—28no—
284C42—22no—
285C42—22no—
286C42—24no—
287C42—282no—
288C42C2×C22⋊D712yes—
289C42C2×C22⋊D712yes—
290C2×C7—21no—
291C2×C7—21no—
292C2×C7C2×#44811yescharacteristic
293C2×C7C2×#45411yescharacteristic
294C2×C7#302711yescharacteristic
295D7—162no—
296C23—284no—
297C23—284no—
298C23—282no—
299C23—282no—
300C23—282no—
301C23C22×C22⋊D711yescharacteristic
302Q8—28no—
303Q8—44no—
304Q8—44no—
305Q8—84no—
306D4—288no—
307D4—562no—
308D4—564no—
309D4—562no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—22no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—282no—
319C2×C4—282no—
320C2×C4—282no—
321C2×C4C2×#114314yes—
322C2×C4C2×#114312yes—
323C2×C4#650512yes—
324C2×C4#651611yescharacteristic
325C2×C4#651612yes—
326C2×C4#651611yescharacteristic
327C2×C4C22×C22⋊D712yes—
328C8—284no—
329C7#1840311yescharacteristic, complemented, Sylow
330C22—21no—
331C22—21no—
332C22—564no—
333C22—564no—
334C22—562no—
335C22—562no—
336C22—562no—
337C22C22×#114311yescharacteristic
338C22C2×#650511yescharacteristic
339C22C2×#651611yescharacteristic
340C4—24no—
341C4—21no—
342C4—22no—
343C4—44no—
344C4—42no—
345C4—44no—
346C4—562no—
347C4#8999914yes—
348C4#9001712yes—
349C2—21no—
350C2—21no—
351C2—1122no—
352C2C2×#8999911yescharacteristic
353C2C2×#9001711yescharacteristic
354C2#1174157111yescharacteristic
subgroupG/H
centerC22C2×#6516
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#3502C2
socleC22×C7C2×#120
prime radicalC2×#6437C22
hypercenter#3502D7
hypocenterC7#18403
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3502

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C7×#3502C2semidirect
4. #11729656C2semidirect
5. #11729589C2semidirect
6. #11729378C2semidirect
74. #3502D7semidirect
85. C2×C7×Q8SD16semidirect
249. C2×Q8#1143semidirect
329. C7#18403semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C7×#35022
#117408002
#117399212
#117296562
#117295892
#117293782
#184037